High-Order Mesh r-Adaptivity with Tangential Relaxation and Guaranteed Mesh Validity

2026-01-25

AI Summary

This work addresses the challenge that high-order meshes in r-adaptivity over curved domains often lose effectiveness due to inadequate handling of tangential degrees of freedom and difficulties in ensuring a strictly positive Jacobian determinant. Building upon the Target-Matrix Optimization Paradigm (TMOP), the authors propose a novel tangential relaxation mechanism that operates without requiring CAD geometry, relying solely on a discrete mesh representation to optimize tangential motion. The method rigorously enforces positivity of the Jacobian determinant throughout the entire high-order r-adaptivity process. To the best of the authors’ knowledge, this is the first approach capable of robustly generating valid, high-quality high-order meshes without access to underlying geometric information, while remaining compatible with arbitrary integration schemes—thereby significantly enhancing simulation reliability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

High-order meshes are crucial for achieving optimal convergence rates in curvilinear domains, preserving symmetry, and aligning with key flow features in moving mesh simulations, but their quality is challenging to control. In prior work, we have developed techniques based on Target-Matrix Optimization Paradigm (TMOP) to adapt a given high-order mesh to the geometry and solution of the partial differential equation (PDE). Here, we extend this framework to address two key gaps in the literature for high-order mesh r-adaptivity. First, we introduce tangential relaxation on curved surfaces using solely the discrete mesh representation, eliminating the need for access to underlying geometry (e.g., CAD model). Second, we ensure a continuously positive Jacobian determinant throughout the domain. This determinant positivity is essential for using the high-order mesh resulting from r-adaptivity with arbitrary quadrature schemes in simulations. The proposed approach is demonstrated to be robust using a variety of numerical experiments.

Problem

• high-order mesh
• r-adaptivity
• tangential relaxation
• mesh validity
• Jacobian determinant

Innovation

• high-order mesh
• r-adaptivity
• tangential relaxation
• mesh validity
• Jacobian positivity